Properties

Label 1-18e2-324.43-r1-0-0
Degree $1$
Conductor $324$
Sign $0.0193 + 0.999i$
Analytic cond. $34.8186$
Root an. cond. $34.8186$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.686 + 0.727i)5-s + (0.993 + 0.116i)7-s + (0.286 − 0.957i)11-s + (−0.0581 + 0.998i)13-s + (−0.939 + 0.342i)17-s + (0.939 + 0.342i)19-s + (0.993 − 0.116i)23-s + (−0.0581 − 0.998i)25-s + (0.893 + 0.448i)29-s + (−0.597 − 0.802i)31-s + (−0.766 + 0.642i)35-s + (0.766 + 0.642i)37-s + (−0.835 − 0.549i)41-s + (−0.973 + 0.230i)43-s + (−0.597 + 0.802i)47-s + ⋯
L(s)  = 1  + (−0.686 + 0.727i)5-s + (0.993 + 0.116i)7-s + (0.286 − 0.957i)11-s + (−0.0581 + 0.998i)13-s + (−0.939 + 0.342i)17-s + (0.939 + 0.342i)19-s + (0.993 − 0.116i)23-s + (−0.0581 − 0.998i)25-s + (0.893 + 0.448i)29-s + (−0.597 − 0.802i)31-s + (−0.766 + 0.642i)35-s + (0.766 + 0.642i)37-s + (−0.835 − 0.549i)41-s + (−0.973 + 0.230i)43-s + (−0.597 + 0.802i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0193 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0193 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.0193 + 0.999i$
Analytic conductor: \(34.8186\)
Root analytic conductor: \(34.8186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 324,\ (1:\ ),\ 0.0193 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.191419356 + 1.168535867i\)
\(L(\frac12)\) \(\approx\) \(1.191419356 + 1.168535867i\)
\(L(1)\) \(\approx\) \(1.040625659 + 0.2711939602i\)
\(L(1)\) \(\approx\) \(1.040625659 + 0.2711939602i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.686 + 0.727i)T \)
7 \( 1 + (0.993 + 0.116i)T \)
11 \( 1 + (0.286 - 0.957i)T \)
13 \( 1 + (-0.0581 + 0.998i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 + (0.993 - 0.116i)T \)
29 \( 1 + (0.893 + 0.448i)T \)
31 \( 1 + (-0.597 - 0.802i)T \)
37 \( 1 + (0.766 + 0.642i)T \)
41 \( 1 + (-0.835 - 0.549i)T \)
43 \( 1 + (-0.973 + 0.230i)T \)
47 \( 1 + (-0.597 + 0.802i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (0.286 + 0.957i)T \)
61 \( 1 + (0.396 + 0.918i)T \)
67 \( 1 + (-0.893 + 0.448i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + (0.173 + 0.984i)T \)
79 \( 1 + (0.835 - 0.549i)T \)
83 \( 1 + (0.835 - 0.549i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (-0.686 - 0.727i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.83610887791508605585371847037, −23.71109205841092682205193657332, −23.05881814176112649159315041294, −22.03567141277666241684636934163, −20.85614766093100856278732109408, −20.17903651731709568111833224638, −19.62796882022201722252360553726, −18.07841379553802399826788009255, −17.614474758871136792207457869354, −16.53489343465370141029319377868, −15.43989052553070362602244439162, −14.90662904615211800669721167163, −13.61056588413262215503555332474, −12.66490811237094269885081858551, −11.746384783861113178178798337395, −10.97540403633940112757869140365, −9.678993999312630099115650517524, −8.63035649221489453109662955933, −7.78208241722830021415371592535, −6.88684882383421624085046545255, −5.09117558856754202107281184917, −4.72855329638000400412246250472, −3.34049272095748635600626193938, −1.771708775040638658247876736, −0.54677153120645444311964643518, 1.23020392262778624107273969225, 2.66786752674853466907067651527, 3.851732739500151649979237349457, 4.85823201872176471441596630306, 6.24107828172057464538140607844, 7.1968648400333122809816887309, 8.22594345737485244525535000135, 9.06894825973082494750554846992, 10.54059256785336911694114331744, 11.40718521275114886276017673977, 11.81731650810469523319785575818, 13.39857565724256173039021565187, 14.318445171528137046298025979917, 14.978094299836240715011452886189, 16.02539816498381569951299960013, 16.95395650036602881630082018458, 18.105924697845372323619097522014, 18.78191199772103435532397399394, 19.62825158959377956845003772055, 20.65498998236059500841169756327, 21.71555459722207727573109080512, 22.261539658099132622236297631825, 23.5353025194353151188195411155, 24.0700382493863571686342947511, 24.94341007331947245329333898269

Graph of the $Z$-function along the critical line